Lemma 5.3 (Theorem 9 in [1]}) 
Suppose \(f_0\in H^\alpha [0,1]\)  for \(\alpha >1/2\) . If \(K_\alpha \)  is used in the GP prior, then it holds with \(\mathbb {P}_0^{(n)}\) -probability at least \(1-n^{-10}\)  that
\(\Vert \hat{f}_n-f_0\Vert _2 \lesssim \left(\frac{\log n}{n}\right)^{\frac{\alpha }{2\alpha +1}},\) 
